We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature. A complete Riemannian manifold M of dimension n is called gradient shrinking Ricci soliton if there exists f ∈ C (M) and a constant ρ > 0 such that Rij +∇i∇jf = ρgij , where Rij denotes the Ricci...

Let M = X×Y be the product of two complex manifolds of positive dimensions. In this paper, we prove that there is no complete Kähler metric g on M such that: either (i) the holomorphic bisectional curvature of g is bounded by a negative constant and the Ricci curvature is bounded below by −C(1+r) where r is the distance from a fixed point; or (ii) g has nonpositive sectional curvature and the h...

The aim of this paper is to characterize $3$-dimensional $N(k)$-paracontact metric manifolds satisfying certain curvature conditions. We prove that a $3$-dimensional $N(k)$-paracontact metric manifold $M$ admits a Ricci soliton whose potential vector field is the Reeb vector field $xi$ if and only if the manifold is a paraSasaki-Einstein manifold. Several consequences of this result are discuss...

Consider a Riemannian manifold ( M m ,...

In this article, we obtain sharp estimate of the Ricci curvature of quaternion slant, bi-slant and semi-slant submanifolds in a quaternion space form, in terms of the squared mean curvature.

It is known that the topological entropy for the geodesic flow on a Riemannian manifoldM is bounded if the absolute value of sectional curvature |KM | is bounded. We replace this condition by the condition of Ricci curvature and injectivity radius.

We extend the concept of singular Ricci flow by Kleiner and Lott from 3d compact manifolds to complete with possibly unbounded curvature. As an application generalized flow, we show that for any Riemannian manifold non-negative curvature, there exists a smooth starting it.